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A Survey on Reproducing Kernel Krein Spaces

Aurelian Gheondea

TL;DR

This survey consolidates the theory of reproducing kernel Kreĭn spaces (RKKS) induced by Hermitian operator-valued kernels, with a focus on existence and uniqueness. It develops the Kolmogorov (linearisation) framework, Kreĭn-space embeddings, and induced-space constructions to illuminate when a given Hermitian kernel yields a unique RKKS, including quasi semidefinite and Pontryagin-space cases. The work highlights a universality phenomenon for holomorphic kernels via Szegő-type kernels and the Drury–Arveson space, and it provides detailed criteria and constructions across single and several complex variable domains. Collectively, the results connect indefinite kernel theory with operator models, dilation theory, and holomorphic function spaces, offering a cohesive toolkit for applications in complex analysis and operator theory.

Abstract

This is a survey on reproducing kernel Krein spaces and their interplay with operator valued Hermitian kernels. Existence and uniqueness properties are carefully reviewed. The approach we follow in this survey uses a more abstract but very useful concept of linearization or Kolmogorov decomposition, as well as the underlying concept of Krein space induced by a selfadjoint operator and that of Krein space continuously embedded. The operator range feature of reproducing kernel spaces is emphasized. We include a careful presentation of Hermitian kernels on complex regions that points out a universality property of the Szego kernels with respect to reproducing kernel Krein spaces of holomorphic functions.

A Survey on Reproducing Kernel Krein Spaces

TL;DR

This survey consolidates the theory of reproducing kernel Kreĭn spaces (RKKS) induced by Hermitian operator-valued kernels, with a focus on existence and uniqueness. It develops the Kolmogorov (linearisation) framework, Kreĭn-space embeddings, and induced-space constructions to illuminate when a given Hermitian kernel yields a unique RKKS, including quasi semidefinite and Pontryagin-space cases. The work highlights a universality phenomenon for holomorphic kernels via Szegő-type kernels and the Drury–Arveson space, and it provides detailed criteria and constructions across single and several complex variable domains. Collectively, the results connect indefinite kernel theory with operator models, dilation theory, and holomorphic function spaces, offering a cohesive toolkit for applications in complex analysis and operator theory.

Abstract

This is a survey on reproducing kernel Krein spaces and their interplay with operator valued Hermitian kernels. Existence and uniqueness properties are carefully reviewed. The approach we follow in this survey uses a more abstract but very useful concept of linearization or Kolmogorov decomposition, as well as the underlying concept of Krein space induced by a selfadjoint operator and that of Krein space continuously embedded. The operator range feature of reproducing kernel spaces is emphasized. We include a careful presentation of Hermitian kernels on complex regions that points out a universality property of the Szego kernels with respect to reproducing kernel Krein spaces of holomorphic functions.

Paper Structure

This paper contains 12 sections, 12 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Kreı n Spaces and their Linear Operators
  3. Hermitian Kernels
  4. Some Examples
  5. Quasi Semidefinite Kernels
  6. Induced Kreı n Spaces
  7. Uniqueness of Induced Kreı n Spaces
  8. Existence of Reproducing Kernel Kreı n Spaces
  9. Uniqueness of Reproducing Kernel Kreı n Spaces
  10. Holomorphic Kernels: Single Variable Domains
  11. Holomorphic Kernels: Several Variables Domains
  12. Comments

Key Result

Theorem 3.1

Let ${\mathcal{K}}$ be a Kreı n space of $\mathbf{H}$-valued vector fields on $X$, that is, ${\mathcal{K}}\subseteq{\mathcal{F}}(\mathbf{H})$. For each $x\in X$ consider the linear operator $E(x)\colon {\mathcal{K}}\rightarrow{\mathcal{H}}_x$ of evaluation at $x$, that is, $E(x)f=f(x)$ for all $f\i hence uniquely determined by ${\mathcal{K}}$.

Theorems & Definitions (21)

  • Theorem 3.1
  • Proposition 3.2
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 4.5
  • Example 4.6
  • Theorem 5.1
  • Theorem 5.2
  • ...and 11 more